A Novel Triple-Frequency Cycle Slip Detection and Correction Method for BDS

Abstract

A novel triple-frequency cycle slip detection and correction method for BDS is proposed to detect and correct cycle slips when the ionosphere is active. Firstly, two sets of phase combinations whose sum equals zero are selected, and the corresponding pseudorange coefficients are obtained by the optimization algorithm, which effectively reduces the ionospheric amplification factor. Then, a set of phase combinations whose sum not equal to zero is adopted to reduce the condition number of coefficient matrices, meanwhile, the ionospheric delay variation between the epochs of this combination data is estimated and corrected. Thus, three linear independence sets of geometry-free and ionosphere-free pseudorange-phase combinations are constructed. Finally, the covariance matrix of the combined observations is obtained by employing observation error and ionospheric correct error, accordingly, the search space and objective function of cycle correction are constructed, which improves the correction accuracy. Based on BDS triple-frequency observations, the experiment results show that the proposed method possesses a good cycle slip detection and correction performance under active ionospheric situation.

1 Introduction

Carrier phase observations are always used in precise GNSS applications and accurate carrier phase observation is the key to precise GNSS applications [1]. Due to interruption of the GNSS transmitted signal, low signal-to-noise ratio and some other reasons, a cycle slip presents a sudden jump of phase ambiguity by an integer number [2]. Detection and correction of cycle slips are necessary before the carrier phases are used as high-precision measurements. At present, 3 or more than 3 carrier frequencies are used in GNSS, and it is a common and effective method to construct the multi-frequency linear combination observation for cycle slips detection and correction [2].

BDS has been able to provide triple-frequency signals. Cycle slip detection using multi-frequency BDS observations has been investigated for a long time. Based on the principle of ionospheric amplification factor and noise minimization, Ref. [3] selected the combination of geometric phase combination and pseudorange-phase combination to detect cycle slips. Reference [4] use wide-lane pseudorange-phase combinations with low ionosphere value as the first and second detectable amount, geometry-free phase combination with low ionosphere value as third detectable amount to detect cycle slip. Due to the current method is difficult to detect cycle slip when the ionosphere is active, a new triple-frequency TurboEdit method for weakening the ionosphere influence is proposed in Ref. [5]. Reference [6] selected two geometry-free phase combinations and one geometry-free pseudorange minus phase linear combination to detect and correct cycle slip in real time. In Ref. [7], a systematic study of BDS multi-frequency linear combination is carried out, and different optimal linear combinations are selected for long and short baselines. In Ref. [8], a robust polynomial adaptive algorithm is proposed for the small cycle of BDS GEOs at low elevation angles, which allows identification of such small cycle slips with high reliability. In above reference, Cycle slips detection and correction methods are studied in detail, but the method of Refs. [3,4,5,6] has a large condition number of coefficient matrix. The methods in the above reference don’t fully consider the covariance of observations, which simplifies the search space and cause some correction errors.

In this paper, by choosing the two sets of coefficients in the group $S_{0}$ [7], the corresponding pseudorange coefficients are solved by using the optimization algorithm firstly. A set of coefficients is selected in the group $S_{1}$, meanwhile, the ionospheric delay variation is eliminated. Three sets of linearly independent, geometric-free ionospheric-free pseudorange-phase combinations are formed, which has better cycle slip detection capability. Through the covariance matrix of the combined observations, a corresponding search space is formed, and the correct cycle slip is searched under the influence of the weight coefficient to minimize the residual of the combined observation. Finally, the method is verified by the BDS triple-frequency data.

2 BDS Triple-Frequency Cycle Slip Detection Method

BDS now has the service capabilities in the Asia Pacific region. Currently the satellites in orbits are able to broadcast triple-frequency navigation signals, respectively, B1: 1561.098 MHz, B2: 1207.14 MHz, B3: 1268.52 MHz. Generally, the frequencies are arranged in descending order, which the three frequencies $f_{1}$, $f_{2}$, $f_{3}$ correspond to B1, B3 and B2 respectively.

2.1 Cycle Slip Detections Using Triple-Frequency Geometric-Free Ionospheric-Free Pseudorange-Phase Combinations

The carrier phase observation in any frequency at epoch $t_{0}$ is expressed as:

$$ \lambda_{i} \varphi_{i} (t_{0} ) = \rho (t_{0} ) + t_{r} (t_{0} ) - t^{s} (t_{0} ) + T(t_{0} ) + \lambda_{i} N_{i} (t_{0} ) - \eta_{i} I_{1} (t_{0} ) + \lambda_{i} \varepsilon_{{\varphi_{i} }} (t_{0} ) $$

(1)

where, $\lambda$ is carrier wavelength, $\varphi$ is carrier phase, $\rho$ is satellite-to-receiver geometric distance, $t_{r}$ is receiver clock error, $t^{s}$ is satellite clock error, $T$ is tropospheric delay, $I_{1}$ is ionospheric delay of the B1 carrier phase, $N$ is integer ambiguity, $\eta_{i}$ is ionosphere amplification factor, $\varepsilon_{\varphi }$ is phase noise, subscripts $i$ represent signals of different frequencies. Pseudorange at epoch $t_{0}$ is expressed as:

$$ P_{l} (t_{0} ) = \rho (t_{0} ) + t_{r} (t_{0} ) - t^{s} (t_{0} ) + T(t_{0} ) + \eta_{l} I_{1} (t_{0} ) + \varepsilon_{{P_{l} }} (t_{0} ) $$

(2)

where, $\rho$, $t_{r}$, $t^{s}$, $T$ have the same meaning with (1), $\eta_{l}$ is ionospheric amplification factor, $\varepsilon_{P}$ is pseudorange noise, the subscript $l$ represents different frequencies. Let the combination coefficients of the triple-frequency carrier phase be $i$, $j$, $k$ and the combination coefficients of the triple-frequency pseudorange be $l$, $m$, $n$ ($l, m,n \in R$, $l + m + n = 1$). The combination of the phase and pseudo-range of the triple-frequency is expressed as:

$$ \begin{aligned} \lambda_{ijk} \varphi_{ijk} & = \lambda_{ijk} (i\varphi_{1} + j\varphi_{2} + k\varphi_{3} ) \\ & = \rho + T + \lambda_{ijk} N_{ijk} - \eta_{ijk} I_{1} + T + (\delta_{s} - \delta_{r} ) + \lambda_{ijk} \varepsilon_{ijk} \\ \end{aligned} $$

(3)

$$ \begin{aligned} P_{lmn} & = lP_{1} + mP_{2} + nP_{3} \\ & = \rho + T + \eta_{lmn} I_{1} + T + (\delta_{s} - \delta_{r} ) + \varepsilon_{lmn} \\ \end{aligned} $$

(4)

where, the ionospheric amplification factors are:

$$ \eta_{ijk} = \frac{{\lambda_{ijk} }}{{\lambda_{1} }}\left( {i + j\frac{{\lambda_{2} }}{{\lambda_{1} }} + k\frac{{\lambda_{3} }}{{\lambda_{1} }}} \right) $$

(5)

$$ \eta_{lmn} = l + m\left( {\frac{{\lambda_{2} }}{{\lambda_{1} }}} \right)^{2} + n\left( {\frac{{\lambda_{3} }}{{\lambda_{1} }}} \right)^{2} $$

(6)

Here, the first-order ionospheric delay terms are mainly discussed. The influences of second-order terms and subsequent terms have little effect on the final result and can be ignored [9]. Differencing (3) and (4), the combined ambiguity of pseudorange-phase can be expressed as:

$$ N_{lijk,lmn} = \varphi_{ijk} - \frac{{P_{lmn} }}{{\lambda_{ijk} }} + \eta_{ijk,lmn} I_{1} + \varepsilon $$

(7)

$$ \eta_{ijk,lmn} = {{(\eta_{ijk} + \eta_{lmn} )} \mathord{\left/ {\vphantom {{(\eta_{ijk} + \eta_{lmn} )} {\lambda_{ijk} }}} \right. \kern-0pt} {\lambda_{ijk} }} $$

(8)

where, $N_{ijk,lmn}$ is integer ambiguity, $\eta_{ijk,lmn}$ is ionospheric amplification factor and $\varepsilon$ is noise. Difference (7) can be obtained difference of ambiguities between adjacent epochs, that is, the cycle slip detection value:

$$ \Delta N_{ijk,lmn} = \Delta \phi_{ijk} - \frac{{\Delta P_{lmn} }}{{\lambda_{ijk} }} + \eta_{ijk,lmn} \Delta I_{1} + \Delta \varepsilon^{\prime} $$

(9)

Assuming that the standard deviation of the carrier phase noise of triple-frequency is same as $\sigma_{\phi }$ and the standard deviation of the pseudorange noise at the triple-frequency is same as $\sigma_{P}$, then the standard deviation of ambiguity is (9):

$$ \sigma_{{\Delta N_{lmn,ijk} }} = \sqrt 2 \sqrt {(i^{2} + j^{2} + k^{2} )\sigma_{\phi }^{2} + {{(l^{2} + m^{2} + n^{2} )\sigma_{P}^{2} } \mathord{\left/ {\vphantom {{(l^{2} + m^{2} + n^{2} )\sigma_{P}^{2} } {\lambda_{ijk}^{2} }}} \right. \kern-0pt} {\lambda_{ijk}^{2} }}} $$

(10)

Make three times the standard deviation of ambiguity estimation (confidence level of 99.7%) as the threshold of cycle slip judgment, and when the following formula holds, it is determined that cycle slip has occured.

$$ i\Delta \phi_{1} + j\Delta \phi_{2} + k\Delta \phi_{3} - \frac{{l\Delta P_{1} + m\Delta P_{2} + n\Delta P_{3} }}{{\lambda_{ijk} }} > 3\sigma_{{\Delta N_{lmn,ijk} }} $$

(11)

2.2 Selection of Triple-Frequency Geometric-Free and Ionosphere-Free Pseudorange-Phase Coefficient

From (7), (9), (11), when the change of $\Delta I_{1}$ is large and the ionospheric amplification factor is large, the ionospheric amplification factor $\eta_{ijk,lmn}$ affect the result of the cycle slip detection. The standard deviation of the ambiguity estkimation $\sigma_{{\Delta N_{lmn,ijk} }}$ also has an impact on the accuracy of cycle slip detection. Therefore, it is necessary to select the appropriate parameters for pseudorange-phase combination to reduce the standard deviation of ambiguity estimation and the ionospheric amplification factor.

2.2.1 Method of Selecting Pseudorange-Phase Coefficient

From (5), (6), (8), the ionospheric amplification factor can be expressed as:

$$ \eta_{ijk,lmn} = i\frac{1}{{\lambda_{1} }} + j\frac{{\lambda_{2} }}{{\lambda_{1}^{2} }} + k\frac{{\lambda_{3} }}{{\lambda_{1}^{2} }} + l\frac{1}{{\lambda_{ijk} }} + m\left( {\frac{{\lambda_{2} }}{{\lambda_{1} }}} \right)^{2} \frac{1}{{\lambda_{ijk} }} + n\left( {\frac{{\lambda_{3} }}{{\lambda_{1} }}} \right)^{2} \frac{1}{{\lambda_{ijk} }} $$

(12)

As can be seen from (10), when the phase coefficient is fixed, the smaller $l^{2} + m^{2} + n^{2}$ is, the smaller standard deviation of the ambiguity estimation. From the inequality $l^{2} + m^{2} + n^{2} \ge 3lmn$ and $l + m + n = 1$, we can see when $l = m = n = {1/3}$, $l^{2} + m^{2} + n^{2}$ has the minimum value, so let $l = m = n = {1/3}$, we can get the smallest $\sigma_{{\Delta N_{lmn,ijk} }}$.

Although when $l = m = n = {1/3}$, $\sigma_{{\Delta N_{lmn,ijk} }}$ has the minimum value, most of the references use this set of pseudorange coefficients. By analysing the Eq. (10) and the cycle slip correction processing, the influence of the ionosphere on the detection results is considerable. If the ionospheric delay can’t be eliminated, the detection and correction of cycle slip will be affected. While the standard deviation of ambiguity estimation can be accurately corrected by search algorithm as long as the deviation is within a certain range. Therefore, the following optimization of pseudorange coefficients is given to further eliminate the ionospheric delay.

$$ \begin{array}{*{20}l} {\mathop {\hbox{min} }\limits_{l,m,n} \left\| {\eta_{ijk,lmn} } \right\|_{2}^{2} } \hfill \\ {s.t.\quad l + m + n = 1\quad \sigma_{{\Delta N_{lmn,ijk} }} < \sigma_{{\Delta N_{\hbox{min} } }} + \omega } \hfill \\ \end{array} $$

(13)

where, the phase coefficient is fixed and $\sigma_{\Delta N\_\hbox{min} }$ is the result when $l = m = n = {1/3}$. $\omega$ is used to limit the standard deviation of the ambiguity estimation, and being adjusted according to the pseudorange noise, the general empirical value is 0.001–0.01.

According to the sum of the coefficients, the phase combination coefficients are classified [7]. Let $S = i + j + k$, $S_{x}$ represent the combination of $i + j + k = \pm x$. In the following table, the appropriate phase combination coefficients are chosen in the group $S_{0}$ and $S_{1}$, and the corresponding pseudorange coefficients are calculated by using the above-mentioned pseudorange coefficients optimization algorithm, compared with the pseudorange coefficients given in [5] and [10], and also compared with the case $l = m = n = {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}$. The standard deviations of the ambiguity estimation and ionospheric amplification factor are mainly compared. The subscript 1 represents the optimization method in this paper, the subscript 2 represents the method in the Refs. [5, 10], and the subscript 3 represents the case when $l = m = n = {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}$. The pseudorange noise is 0.3 m and the phase noise is 0.01 cycle.

It can be seen from Table 1 that the proposed method can further reduce the ionospheric amplification factor. In the above table, the coefficients (0, 1, −1) and (1, −2, 1) are selected as the two sets of cycle slip detection combinations. In order to constitute three linear independent equations and reduce the condition number of the coefficient matrix, another set of phase coefficients are selected in the $S_{1}$ group. Subscript 1 represents the optimization method proposed in this paper, and subscript 2 is the case when $l = m = n = {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}$.

Table 1 The optimal combination in $S_{0}$ region

It can be seen from Table 2 that although the optimized pseudorange coefficients can greatly reduce the ionospheric amplification factor, the standard deviation of the ambiguity estimation becomes very large. Therefore, the set of $l = m = n = {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}$ is used for the $S_{1}$ group. (−4, 3, 2) is selected from $S_{1}$ group. This set of coefficient has longer wavelength, the standard deviation of ambiguity estimation is smaller. However, the ionospheric amplification factor of (−4, 3, 2) is larger, which needs to eliminate the influence of ionospheric delay.

Table 2 The optimal combination in $S_{1}$ region

2.2.2 Method of Eliminating Ionospheric Delay of ${\mathbf{S}}_{1}$

To eliminate the ionospheric delay, we need to estimate the ionospheric delay variation between adjacent epochs at the B1 frequency. We can calculate $\Delta I_{1}$ and its standard deviation using Eq. (14). There is no cycle slip in the phase observation [10].

$$ \Delta I_{1} = \frac{{\lambda_{i} \Delta \varphi_{i} - \lambda_{j} \Delta \varphi_{j} }}{{\eta_{j} - \eta_{i} }} , \sigma ( \Delta I_{1} ) = \frac{{2\sigma_{\varphi } }}{{\left| {\eta_{j} - \eta_{i} } \right|}} $$

(14)

When there are cycle slips in the phase observation, there is a difference $\Delta N$ between epochs. $\Delta N_{i}$ and $\Delta N_{j}$ between the frequencies can’t offset each other, which have a greater impact on the $\Delta I_{1}$ estimation. When three frequency exist the same cycle slip $\Delta N$, the first two sets of coefficients can’t detect the cycle slip. The ionospheric quadratic difference defined by (15) can be used to detect it. Due to the second epoch difference of the ionospheric delay is very smooth, if there is a large deviation, we can consider triple-frequency exist the same cycle slip.

$$ \nabla \Delta I_{1} (\alpha ) = \Delta I_{1} (\alpha ) - \Delta I_{1} (\alpha - 1) $$

(15)

When any one of $\Delta N_{(0, 1, -1)}$, $\Delta N_{(1, -2, 1)}$, and $\nabla \Delta I_{1} (\alpha )$ exceeds threshold, it is determined that cycle slips have occurred. Therefore, $\Delta I_{1}$ of the current epoch can’t be calculated using Eq. (14) and can be corrected by the following equation.

$$ \Delta I_{1} (\alpha ) = \Delta I_{1} (\alpha - 1) + \nabla \Delta \bar{I}_{1} (\alpha - 1) $$

(16)

where $\nabla \Delta \bar{I}_{1}$ is the smoothed value of $\nabla \Delta I_{1}$ without cycle slips. The calculation equation is expressed as:

$$ \nabla \Delta \bar{I}_{1} (\alpha ) = \theta \cdot \nabla \Delta I_{1} (\alpha ) + (1 - \theta ) \cdot \nabla \Delta I_{1} (\alpha - 1) $$

(17)

$\theta$ is the weight factor. The ionospheric delay of the $S_{1}$ group coefficients (−4, 3, 2) can be corrected by using the calculated $\Delta I_{1}$.

3 A New Cycle Slip Correction Method

Three pseudorange-phase combinations are used to construct cycle slip detection with geometry-free and ionosphere-free, and the above method is used to correct the ionospheric delay of $S_{1}$ group observations. The combined observations can be obtained from:

$$ {\mathbf{L = CL^{\prime}}} $$

(18)

$$ {\mathbf{C}} = \left[ {\begin{array}{*{20}c} {i_{1} } & {j_{1} } & {k_{1} } & {\frac{{ - l_{1} }}{{\lambda_{{i_{1} j_{1} k_{1} }} }}} & {\frac{{ - m_{1} }}{{\lambda_{{i_{1} j_{1} k_{1} }} }}} & {\frac{{ - n_{1} }}{{\lambda_{{i_{1} j_{1} k_{1} }} }}} \\ {i_{2} } & {j_{2} } & {k_{2} } & {\frac{{ - l_{2} }}{{\lambda_{{i_{2} j_{2} k_{2} }} }}} & {\frac{{ - m_{2} }}{{\lambda_{{i_{2} j_{2} k_{2} }} }}} & {\frac{{ - n_{2} }}{{\lambda_{{i_{2} j_{2} k_{2} }} }}} \\ {i_{3} } & {j_{3} } & {k_{3} } & {\frac{{ - l_{3} }}{{\lambda_{{i_{3} j_{3} k_{3} }} }}} & {\frac{{ - m_{3} }}{{\lambda_{{i_{3} j_{3} k_{3} }} }}} & {\frac{{ - n_{3} }}{{\lambda_{{i_{3} j_{3} k_{3} }} }}} \\ \end{array} } \right] $$

(19)

$$ {\mathbf{L}}^{\prime} = \left[ {\begin{array}{*{20}c} {l_{{\varphi_{1} }} } & {l_{{\varphi_{2} }} } & {l_{{\varphi_{3} }} } & {l_{{P_{1} }} } & {l_{{P_{2} }} } & {l_{{P_{3} }} } \\ \end{array} } \right]^{T} $$

(20)

where, the phase coefficients $(i_{1} ,j_{1} ,k_{1} )$ and $(i_{2} ,j_{2} ,k_{2} )$ correspond to (0, 1, −1) and (1, −2, 1) in $S_{0}$ group, and the corresponding pseudorange coefficients are in Table 1. In this equation, The phase coefficient $(i_{3} ,j_{3} ,k_{3} )$ corresponds to (−4, 3, 2) in $S_{1}$ group and the pseudorange coefficient is $l = m = n = {1/3}$. $l_{{\varphi_{i} }}$ and $l_{{P_{i} }}$ are phase and pseudorange observations in ${\mathbf{L}}^{\prime }$ respectively. The relationship between the combination coefficient and the cycle slip valuation is expressed as:

$$ {\mathbf{AX}} = {\mathbf{L}} $$

(21)

$$ {\mathbf{A}} = \left[ {\begin{array}{*{20}c} {i_{1} } & {j_{1} } & {k_{1} } \\ {i_{2} } & {j_{2} } & {k_{2} } \\ {i_{3} } & {j_{3} } & {k_{3} } \\ \end{array} } \right]{\mathbf{X}} = \left[ {\begin{array}{*{20}c} {\Delta N_{1} } & {\Delta N_{2} } & {\Delta N_{3} } \\ \end{array} } \right]^{T} $$

(22)

where, the coefficient matrix ${\mathbf{A}}$ is reversible, and the cycle slip can be determined by ${\mathbf{X}} = {\mathbf{A}}^{ - 1} {\mathbf{L}}$. But for the first and second observations, in order to reduce ionospheric amplification factor, the standard deviation of ambiguity estimation is increased. Ambiguity estimation of the third detection amount is affected by the error of ionospheric delay estimation. Therefore, it is not easy to correct cycle slip by calculating ${\mathbf{X}} = {\mathbf{A}}^{ - 1} {\mathbf{L}}$ and rounding.

Assuming that the covariance matrix of the combined observation ${\mathbf{L}}$ is ${\mathbf{Q}}$ and the third combined observation is also affected by the variance of the ionospheric delay estimation, we consider it into the covariance matrix, then ${\mathbf{Q}}$ is defined as follows:

$$ {\mathbf{Q}} = [{\mathbf{C}} \, {\mathbf{e}}_{3} ]{\mathbf{Q}}_{0} [{\mathbf{C}} \, {\mathbf{e}}_{3} ]^{T} $$

(23)

where ${\mathbf{Q}}_{0} = diag(\sigma_{{\varphi_{1} }}^{2} ,\sigma_{{\varphi_{2} }}^{2} ,\sigma_{{\varphi_{3} }}^{2} ,\sigma_{{P_{1} }}^{2} ,\sigma_{{P_{2} }}^{2} ,\sigma_{{P_{3} }}^{2} ,\sigma_{{\Delta I_{1} }}^{2} )$ ${\mathbf{e}}_{3} = \left[ {\begin{array}{*{20}c} 0 & 0 & 1 \\ \end{array} } \right]^{T}$, $\sigma_{{\varphi_{i} }}^{{}}$, $\sigma_{{P_{i} }}^{{}}$, $\sigma_{{\Delta I_{1} }}^{{}}$ denote the standard deviation of phase and pseudorange and ionospheric correct error respectively. In order to determine the integer cycle slip ${\mathbf{X}}_{N}$, the corresponding search space is set up as:

$$ ({\mathbf{X}}_{N} - {\hat{\mathbf{X}}})^{T} {\mathbf{Q}}^{ - 1} ({\mathbf{X}}_{N} - {\hat{\mathbf{X}}}) < \chi $$

(24)

The threshold $\chi$ is generally based on the observation noise. The target equation is expressed as:

$$ ({\mathbf{AX}}_{N} - {\mathbf{L}})^{T} {\mathbf{Q}}^{ - 1} ({\mathbf{AX}}_{N} - {\mathbf{L}}) = \hbox{min} $$

(25)

Under the influence of weight coefficient, the correct cycle slip minimizes the residuals of ${\mathbf{AX}}_{N}$ and combined observations ${\mathbf{L}}$.

4 Experimental Analysis

The experiment adopts the data in JFNG observation station ($30^\circ 31^{\prime } N,114^\circ 29^{\prime } E$) at a large magnetic storm on December 21, 2015, the satellite elevation cutoff angle is set to $15^{ \circ }$. The geomagnetic Dst index from 0 to 6 h on that day was abnormal, indicating that a large magnetic storm occurred. Using the above method, the C04 satellite data (without cycle slips) are processed. As can be seen from Fig. 1, the large magnetic storms do not affect the detection value due to the elimination of the ionospheric effects.

Fig. 1

Detection of three groups when no cycle slips occur

Figure 2 shows the results of four sets of special cycle slips. The special cycle slips include the cycle slips which are not sensitive to each set of coefficients and the small jumps which occur simultaneously in triple-frequency. As can be seen from Fig. 2, when a group of coefficients is not sensitive to the cycle slips, other coefficients can detect cycle slip. For $\Delta N_{1} = \Delta N_{2} = \Delta N_{3} = 1$ this type of cycle slip, coefficients in $S_{1}$ group can be detected.

Fig. 2

Detection of three groups when special cycle slips occur

In order to further verify the performance of this cycle slip detection and correction method, different cycle slips combinations are added in different epochs of carrier phase observations of C08 and C12 satellites, including small cycle slips, large cycle slips and special cycle slips. As can be seen from Table 3, this method eliminates the effect of the ionospheric delay between epochs and can detect and correct the cycle slip. When some cycle slips occur, it is not correct to use the ${\mathbf{X}} = {\mathbf{A}}^{ - 1} {\mathbf{L}}$ directly to get the cycle slips. Using the method mentioned in this paper to search the optimal value in a certain region can avoid the problems caused by pseudorange, phase and ionospheric repair errors.

Table 3 Cycle slips detection and correct results

5 Conclusions

A new triple-frequency cycle slip detection and correction method for BDS is proposed when the ionosphere is active. Firstly, two sets of phase combinations are selected from group $S_{0}$, and the corresponding pseudorange coefficients are obtained by the optimization algorithm, which effectively reduces the ionospheric amplification factor. Then, a set of phase combinations is selected from group $S_{1}$ to reduce the condition number of coefficient matrices, meanwhile, the ionospheric delay variation between the epochs of this combination data is estimated and corrected. Thus, three linear independence sets of geometry-free and ionosphere-free pseudorange-phase combinations are constructed. Finally, the covariance matrix of the combined observations is obtained by using observation and ionospheric correct error, which is employed to establish searching space. Under the influence of weight coefficient, the correct cycle slip is searched to minimize the residual of the combined observation. At last, the performance of proposed method is verified by BDS triple-frequency data. When the ionosphere is active, this method possesses a good cycle slip detection and correction performance.